1. AKCE Int. J. Graphs Comb., X, No. X (XXXX), pp. 1-12 AN UPPER BOUND FOR FUZZY CHROMATIC NUMBER OF FUZZY GRAPHS AND THEIR COMPLEMENT Isnaini Rosyida Ph.D Student at Department of Mathematics, GadjahMada University, Indonesia Department of Mathematics, Semarang State University, Indonesia e-mail: iisisnaini@gmail.com S.Lavanya Department of Mathematics, D.G. Vaishnav College, Chennai, Tamilnadu, India e-mail: lavanyaprasad1@gmail.com Widodo, Ch.R.Indrati Department of Mathematics, GadjahMada University, Indonesia K.A.Sugeng Department of Mathematics, University of Indonesia, Indonesia Abstract An upper bound for the sum and product of chromatic number of a crisp graph and chromatic number of its complement has been given by Nordhaus and Gaddum. On the other hand, Sattanathan and Lavanya gave a theorem about un upper bound for the sum and product of fuzzy chromatic number of a fuzzy graph and its complement. In this paper, we give an example that the bounds given by Sattanathan and Lavanya do not hold for certain fuzzy graphs. Further, we add conditions to a fuzzy graph so it satisﬁes the theorem of Sattanathan and LavanyaKeywords: Chromatic Number, Fuzzy Coloring, Fuzzy graph, Fuzzy Chromatic Num-ber, Complement Fuzzy graph 2000 Mathematics Subject Classiﬁcation: 1. Introduction Let G be a graph with vertex set V(G) and edge set E(G). Vertex coloring of G is amapping C : V (G) → N with N is a set of natural numbers, such that C(x) = C(y)if (x, y) ∈ E(G) . Given an integer k , a k -coloring of G is a mapping C : V (G) →{1, 2, ..., k} such that C(x) = C(y) if (x, y) ∈ E(G) . The chromatic number of G ,denoted by χ(G) is the smallest integer k such that there is a k -coloring of G . Forsimplicity, we will use symbol V for V (G) and E for E(G) .
2. 2 An Upper of fuzzy chromatic number of Fuzzy graphs and their Complement Vertex coloring of graph G can be interpreted as a problem of special kind of partitionof the vertex set as in [1]. Therefore there is another deﬁnition of vertex coloring as follows.A color partition of general graph G(V, E) is a partition of V into subsets, called color-classes, such that V = V1 ∪ V2 ∪ ... ∪ Vk where the subsets Vi ( 1 ≤ i ≤ k ) are non-emptyand mutually disjoint, and each Vi contains no pair of adjacent vertices. The chromaticnumber of G is the smallest natural number k for which such a partition is possible. The Fuzzy set theory has been introduced by Zadeh in 1965 [10]. Ideas of fuzzy settheory have been introduced into graph theory by Rosenfeld in 1975 as in [5]. We calla graph G(V, E) by crisp graph. Rosenfeld has expanded the crisp graph G(V, E) into ˜fuzzy graph G(V, σ, µ) that is a graph which has fuzzy vertex set with the membershipfunction σ : V → [0, 1] and fuzzy edge set with the membership function µ : E → [0, 1] . ˜While Kaufman has introduced a fuzzy graph G(V, σ, µ) that is a graph with crisp vertexset and fuzzy edge set as in [6]. ˜ The vertex coloring of a fuzzy graph G(V, σ, µ) was introduced by Munoz et al [6].Munoz expanded the concept of coloring function C : V → N of a crisp graph into coloringfunction Cd,f : V → S of a fuzzy graph, where S is the available color set, d is a distancefunction deﬁned between the colors on S, and f is a real scale function deﬁned on image( µ ). Pourpasha and Soheilifar [7] expanded the coloring function C d,f : V → S of fuzzy ˜ ˜graph G(V, µ) into the coloring function Cd,f,g : V → S of a fuzzy graph G(V, σ, µ)where g is a real scale function deﬁned on image ( σ ). Further, the fuzzy vertex coloring ˜of a fuzzy graph G(V, σ, µ) was deﬁned by Eslahchi and Onagh [2]. They deﬁned fuzzyvertex coloring of a fuzzy graph through fuzzy color-classes of V . Several authors have studied the problem of obtaining an upper bound for the chromaticnumber of complementary crisp graphs and complementary fuzzy graphs. According toSattanathan and Lavanya as in [8], Nordhaus and Gaddum gave the ﬁrst theorem aboutan upper bound for crisp graph and its complement in 1956. While Sattanathan and La-vanya [8] gave an upper bound for the sum and product of the fuzzy chromatic number ofcomplementary fuzzy graphs. Moreover, Lavanya and Sattanathan [4] gave the deﬁnitionof fuzzy vertex coloring of fuzzy graph with slight modiﬁcations. In this paper we give anexample of fuzzy graphs which do not satisfy the theorem of Sattanathan and Lavanya.We also give some theorems on fuzzy chromatic number of fuzzy cycle and its comple-ment. Further, we add conditions to a fuzzy graph such that it satisﬁes the theorem ofSattanathan and Lavanya. 2. Preliminaries We review brieﬂy some deﬁnitions in fuzzy sets as in [10]. Let X be a space of objects.A fuzzy set A on X is the set of the form {(x, µ A (x)): x ∈ X }, where µA is a mapping:X → [0, 1] . We call µA is a membership function of the fuzzy set A , and µ A (x) at xrepresenting the grade of membership of x in A . Other deﬁnition said that a fuzzy set A on X is a mapping µ : X → [0, 1] , as in [3].According to the ﬁrst notation, the symbol of the fuzzy set A is distinguished from the
3. I.Rosyida, S.Lavanya, Widodo, Ch.R.Indrati, K.A.Sugeng 3symbol of its membership function (µ A ) . According to the second notation, there is nodistinction between the two symbols. In this paper we use the second notation. A fuzzy set µ on X is empty if and only if µ(x) = 0 for all x ∈ X . Let µ and σbe fuzzy sets on X. The union µ ∪ σ is the fuzzy set on X deﬁned by (µ ∪ σ)(x) =max{µ(x), σ(x)} for all x ∈ X . The intersection µ ∩ σ is the fuzzy set on X deﬁned by(µ ∩ σ)(x) = min{µ(x), σ(x)} for all x ∈ X . We review brieﬂy some deﬁnitions in fuzzy graphs as in [5], and [9].Deﬁnition 2.1. ([5]) Let V be a ﬁnite nonempty set. Let E ⊆ V × V . A fuzzy graph˜G(V, σ, µ) is a graph consisting of a pair of functions (σ, µ) where σ is a fuzzy set onV and µ is a fuzzy set on E, i.e. σ : V → [0, 1] and µ : E → [0, 1] such thatµ(u, v) ≤ min{σ(u), σ(v)} for all u, v ∈ V .Note that a crisp graph G(V, E) is a special case of a fuzzy graph with each vertex andedge of V and E having degree of membership 1. In this paper we refer the deﬁnition of complement of fuzzy graph as in Sunitha andVijayakumar’s paper [9]. ˜ ¯ ˜Deﬁnition 2.2. ([9]) The complement of a fuzzy graph G(V, σ, µ) is a fuzzy graph G =(¯ , µ) where σ = σ and µ(u, v) = min{σ(u), σ(v)} − µ(u, v) for all u, v ∈ V. σ ¯ ¯ ¯ ˜ ˜ ¯ ˜Deﬁnition 2.3. ([9]) A fuzzy graph G(V, σ, µ) is self complementary if G = G. The deﬁnition of adjacency is classify as strong and weak adjacency as in [4]. ˜Deﬁnition 2.4. ([4]) Two vertices u and v of fuzzy graph G(V, σ, µ) are called strongly 1adjacent if µ(u, v) ≥ 2 min{σ(u), σ(v)} , otherwise weakly adjacent. Sunitha and Vijayakumar in [9] gave the condition for a fuzzy graph to be self comple-mentary as follows. ˜Theorem 2.5. ([9]) Let G(V, σ, µ) be a fuzzy graph. If µ(u, v) = 1 min{σ(u), σ(v)} for 2all u, v ∈ V then G is self complementary. We use the deﬁnition of fuzzy vertex coloring and fuzzy chromatic number as in [4].Deﬁnition 2.6. ([4]) A family Γ = {γ1 , γ2 , ..., γk } of fuzzy subsets on V is called a ˜k -fuzzy coloring of G(V, σ, µ) if a. γ1 ∪ γ2 ∪ ... ∪ γk = σ b. min {γi (u), γj (u) | 1 ≤ i = j ≤ k} = 0 for all u ∈ V ˜ c. for every strongly adjacent vertices u, v of G , min{γi (u), γi (v)} = 0 (1 ≤ i ≤ k).
4. 4 An Upper of fuzzy chromatic number of Fuzzy graphs and their Complement ˜ ˜Fuzzy chromatic number of G is the least value of k for which the fuzzy graph G has ˜k -fuzzy coloring and is denoted by χ F (G). ¯ ˜ ¯ ¯ ˜ Fuzzy chromatic number of the complement G is denoted by χF (G). For simplicity, ˜ ¯ ¯we will use symbol χF for χF (G) and χF for χF (G). ¯ ˜ ˜ In [4], the authors gave the theorem about strongly adjacent vertices in G correspond ¯ ˜to weakly adjacent vertices in G and vice versa. ˜ ¯ ˜Theorem 2.7. ([4]) Let G(V, σ , µ ) be a fuzzy graph and G = (¯ , µ) be its complement. σ ¯ ˜The vertices u, v ∈ V are strongly adjacent in G if and only if u, v are weakly adjacent ¯ ˜in G. The strongly adjacent vertices u, v ∈ V in this theorem is restricted to µ(u, v) >12 min{σ(u), σ(v)}. Next, we give a lemma on the fuzzy chromatic number of a fuzzy graph which everypair of its vertices are weakly adjacent. ˜Lemma 2.8. If G(V, σ, µ) is a fuzzy graph which every pair of vertices are weakly adjacent ˜ = 1.then χF (G)Proof. Since every pair of vertices are weakly adjacent then we can construct the familyΓ which has only one member satisfying all of the conditions in Deﬁnition 2.6. Thus ˜χF (G) = 1. ˜The fuzzy graph G as in Lemma 2.8 is called trivial. 3. Main Results In the following, we give a lemma on the fuzzy chromatic number of a fuzzy graph whichevery pair of its vertices are strongly adjacent ˜Lemma 3.9. If G(V, σ, µ) is a fuzzy graph with n vertices which every pair of vertices ˜are strongly adjacent then χF (G) = nProof. Let V = {v1 , v2 , ..., vn } . Since every pair of vertices are strongly adjacent then wecan construct the family Γ = {γ1 , γ2 , . . . , γn } where σ(xi ), x = xi γi (x) = 0, x = xi we can see that the family Γ satisfy the properties a,b,c in Deﬁnition 2.6. This is theminimal fuzzy coloring since any family with less than n members does not satisfy all of ˜the conditions in Deﬁnition 2.6. Thus χ F (G) = n.
5. I.Rosyida, S.Lavanya, Widodo, Ch.R.Indrati, K.A.Sugeng 53.1. Fuzzy Chromatic Number of a Fuzzy Cycle and Its Complement In crisp graph case, the chromatic number of even cycle is 2 and the chromatic numberof odd cycle is 3. In the case of fuzzy cycle, we have the fuzzy chromatic number of evencycle is 2 and the fuzzy chromatic number of odd cycle is 2 or 3 which is explained in ˜Theorem 3.11 and 3.12. In the results below, the fuzzy graph G is not trivial. This ˜means that G has at least a pair of vertices which are strongly adjacent. First, we give adeﬁnition on fuzzy cycle ˜Deﬁnition 3.10. A cycle C in a fuzzy graph G(V, σ, µ) is a sequences of distinct vertices(u0 , u1 , u2 , ..., un ) such that µ(ui , ui+1 ) > 0 for 1 ≤ i ≤ n and µ(uj , uk ) = 0 for k =j + 1 , where u0 = un and n is called the length of C . A cycle C in a fuzzy graph G ˜can be called by fuzzy cycle. ˜ ˜Theorem 3.11. If G(V, σ, µ) is a fuzzy cycle with even number of vertices then χ F (G) =2. ˜Proof. Let G(V, σ, µ) be a fuzzy cycle with even number of vertices. Let V = ˜ ˜{v1 , v2 , ..., vn } where n is even. Since G is not trivial then χF (G) > 1. Further, since G is˜a fuzzy cycle then µ(ui , ui+1 ) > 0 , for 1 ≤ i ≤ n , and µ(uj , uk ) = 0 for k = j + 1 . Since1 12 min{σ(uj ), σ(uk )} = 0 then µ(uj , uk ) < 2 min{σ(uj ), σ(uk )} . This means that all ofthe pair of vertices (uj , uk ) with k = j + 1 are weakly adjacent. So that the vertex setV can be partitioned into two sets V1 = {v1 , v3 , v5 , ..., vn−1 } and V2 = {v2 , v4 , v6 , ..., vn }where every pair of vertices in Vi are weakly adjacent. Further, we can construct thefamily Γ = {γ1 , γ2 } where σ(vi ), if i is odd, i = 1, 3, 5, ..., n − 1 γ1 (vi ) = 0, if i is even, i = 2, 4, 6, ..., n 0, if i is odd γ2 (vi ) = σ(vi ), if i is even.Here we can see that the family Γ satisfy all of the conditions in Deﬁnition 2.6. This isthe minimal fuzzy coloring since any family with less than 2 members does not satisfy all ˜of the conditions in Deﬁnition 2.6. Thus χ F (G) = 2. ˜Theorem 3.12. If G(V, σ, µ) is a fuzzy cycle with odd number of vertices then χ F (G) = 2 ˜ ˜if there exists at least a pair of vertices (u k , uk+1 ) which are weakly adjacent, and χF (G) =3 if all of the pair of vertices (ui , ui+1 ) are strongly adjacent for all i = 1, 2, ..., n − 1 ˜Proof. Let G(V, σ, µ) be a fuzzy cycle with odd number of vertices. Let V ={v1 , v2 , ..., vn } where n is odd. We consider two cases.
6. 6 An Upper of fuzzy chromatic number of Fuzzy graphs and their ComplementCase 1: There exists at least a pair of vertices (u k , uk+1 ) which are weakly adjacent.Without loss of generality, we assume that v 1 v2 is weakly adjacent. In the same argumentwith the proof of theorem 3.11 the vertex set V can be partitioned into two sets V 1 ={v3 , v5 , ..., vn−1 } and V2 = {v1 , v2 , v4 , ..., vn−2 , vn } where every pair of vertices in Vi areweakly adjacent. Further, we can construct the family Γ = {γ 1 , γ2 } where σ(v), if v ∈ V1 γ1 (v) = 0, if v ∈ V2 . 0, if v ∈ V1 γ2 (v) = σ(v), if v ∈ V2The family Γ satisfy all of the conditions in Deﬁnition 2.6. This is the minimal fuzzycoloring since any family with less than 2 members does not satisfy all of the conditions ˜in Deﬁnition 2.6. Thus χF (G) = 2.Case 2: All of the pair of vertices (ui , ui+1 ) are strongly adjacent for all i = 1, 2, ..., n−1 . ˜Since G has odd number of vertices then the vertex set V can be partitioned into threesets V1 = {v1 , v3 , v5 , ..., vn−1 } , V2 = {v2 , v4 , v6 , ..., vn−2 } and V3 = {vn }. In the sameargument with the proof of Theorem 3.11, every pair of vertices (u j , uk ) with k = j + 1 ˜are weakly adjacent in G. This means that every pair of vertices in V i are weakly adjacent.So that, we can construct the family Γ = {γ 1 , γ2 , γ3 } where σ(v), if v ∈ Vk γk (v) = 0, if v ∈ Vk for all k = 1, 2, 3.The family Γ satisfy all of the conditions in Deﬁnition 2.6. This is the minimal fuzzycoloring since any family with less than 3 members does not satisfy all of the conditions ˜in Deﬁnition 2.6. Thus χF (G) = 3. Next, we give a theorem on an upper bound for the fuzzy chromatic number of fuzzycycles and their complement ˜Theorem 3.13. If G(V, σ, µ) is a fuzzy cycle with n vertices ( n ≥ 4 ), then χF + χF ≤ 2(n − 1) and χF · χF ≤ n(n − 2) ¯ ¯ ˜ ¯ ˜Proof. Let G(V, σ, µ) be a fuzzy cycle with n vertices and G = (¯ , µ) be a complement σ ¯ ˜ Let V = {v1 , v2 , ..., vn }. We consider two cases.of G.Case 1: µ(ui , ui+1 ) = 1 min{σ(ui ), σ(ui+1 )} for all i = 1, 2, . . . , n − 1 . 2
7. I.Rosyida, S.Lavanya, Widodo, Ch.R.Indrati, K.A.Sugeng 7Then µ(ui , ui+1 ) = 1 min{σ(ui ), σ(ui+1 )} for all i = 1, 2, . . . , n − 1 . In other words, the ¯ 2 ¯ ˜pair of vertices (ui , ui+1 ) are strongly adjacent in G. In the same argument with theproof of Theorem 3.11, every pair of vertices (u j , uk ) with k = j + 1 are weakly adjacent ˜ ¯ ˜in G. Then this vertices are strongly adjacent in G . Thus, every pair of vertices arestrongly adjacent in G ¯ . By lemma 3.9, χ = n . ˜ ¯ FBy the Theorem 3.11, if n is even and n ≥ 4 then χF + χF = 2 + n ≤ (n − 2) + n = 2(n − 1) ¯ and χF · χF = 2n ≤ (n − 2)n ¯By the Theorem 3, if n is odd and n ≥ 5 then χF + χF ≤ 3 + n ≤ (n − 2) + n = 2(n − 1) ¯ and χF · χF = 3n ≤ (n − 2)n ¯ ˜Case 2: G has at least a pair of vertices (uk , uk+1 ) such that µ(uk , uk+1 ) =12 min{σ(uk ), σ(uk+1 )}This means that there is at least a pair of vertices u k and uk+1 which are weakly adjacent ¯ ˜ ¯ ˜in G. So that we can construct the family Γ = {γ 1 , γ2 , ..., γk } for k ≤ n − 1 on G where γi (u) = σ(u), and γi (v) = σ(v) ¯ ˜ if u and v are weakly adjacent in G ¯ ˜ γk (u) = σ(u), γk (v) = 0, k = i, if u and v are strongly adjacent in GThe family Γ satisfy all of the conditions in Deﬁnition 2.6. Thus χ F ≤ n − 1 . ¯By theorem 3.11 and 3.12, χF + χF ≤ 3 + (n − 1) ¯ and χF · χF ≤ 3(n − 1) . ¯If n is even and n ≥ 4 then χF + χF ≤ 3 + (n − 1) ≤ (n − 1) + (n − 1) = 2(n − 1) ¯ andχF · χF ≤ 3(n − 1) ≤ (n − 1)(n − 1) ≤ n(n − 2) . ¯If n is odd and n ≥ 4 then χF + χF ≤ 3 + (n − 1) ≤ (n − 2) + (n − 1) ≤ 2(n − 1) ¯ andχF · χF ≤ 3(n − 1) ≤ (n − 2)(n − 1) ≤ n(n − 2) . ¯3.2. Fuzzy Chromatic Number of a Fuzzy Graph and Its Complement In crisp graph case, an upper bound for the chromatic number of a graph and itscomplement has been given by Nordhaus and Gaddum as in [8]. In case of fuzzy graph,an upper bound has been given by Sattanathan and Lavanya [8].Theorem 3.14. (Nordhaus and Gaddum) If G(V, E) is a graph with n vertices then √ ¯ 2 n ≤ χ(G) + χ(G) ≤ n + 1 ¯ n+1 2 n ≤ χ(G) · χ(G) ≤ ( ) 2
8. 8 An Upper of fuzzy chromatic number of Fuzzy graphs and their Complement ˜ ¯ ˜Theorem 3.15. (Sattanathan and Lavanya) If G(V σ, µ) is a fuzzy graph and G = (¯ , µ) σ ¯ ˜ , thenis a complement of G ˜ ¯ ¯ ˜ χF (G) + χF (G) ≤ 2(n − 1) ˜ ¯ ¯ ˜ χF (G) · χF (G) ≤ n2 + 1 In example 3.16 and 3.17 we give two fuzzy graphs which do not satisfy Theorem 3.14and 3.15. ˜Example 3.16. Let G(V, σ, µ) be a fuzzy graph which is self complementary. It meansthat µ(u, v) = µ(u, v) = 1 min{σ(u), σ(v)} for all u, v ∈ V . ¯ 2 A 0.2 0.1 0.1 0.1 B 0.4 D 0.3 0.15 0.2 0.15 C 0.5 ˜ ¯ ˜ Figure 1: A fuzzy graph G = GLet Γ = {γ1 , γ2 , γ3 , γ4 } be a family of sets deﬁned on V as in Table 1. ˜ Table 1: The Family Γ = {γ1 , γ2 , γ3 , γ4 } of fuzzy sets on V of Fuzzy Graph G Vertices γ1 γ2 γ3 γ4 Max A 0.2 0 0 0 0.2 B 0 0.4 0 0 0.4 C 0 0 0.5 0 0.5 D 0 0 0 0.3 0.3Here we can see that the family Γ satisfy all of the conditions in Deﬁnition 2.6. Thisis the minimal fuzzy coloring since any family with less than 4 members does not satisfy ˜ ˜the conditions in Deﬁnition 2.6. Thus χ F (G) = 4 . Since G is self complementary then¯ ˜ ¯ = 4. The sum of the fuzzy chromatic number of G and the fuzzy chromatic numberχF (G) ˜of G¯ as follows: χ (G) + χ (G) = 8 > 6 = 2(n − 1). ˜ ˜ ¯F ˜ ¯ F ˜Thus G does not satisfy Theorem 3.15.
9. I.Rosyida, S.Lavanya, Widodo, Ch.R.Indrati, K.A.Sugeng 9 We also have other example of fuzzy graph which it does not satisfy Theorem 3.15. Thefuzzy graph can be seen in the following example. ˜Example 3.17. Let G(V, σ, µ) is a fuzzy graph as in the ﬁgure below. A 0.3 A 0.3 0.15 0.15 0.15 0.15 0.15 0.15 B 0.4 D 0.6 B 0.4 D 0.6 0.2 0.2 0.2 0.2 0.2 0.3 C 0.5 C 0.5 ˜ ¯ ˜ Figure 2: A fuzzy graph G and its complement G ˜We can see that CD is weakly adjacent in G . while vertex A is strongly adjacent toB and D . Then the family satisfying all of the conditions in deﬁnition of fuzzy vertex ˜coloring of G is given by Table 2. ˜ Table 2: The Family Γ = {γ1 , γ2 , γ3 } of Fuzzy Graph G in the ﬁgure 2 Vertices γ1 γ2 γ3 Max A 0.3 0 0 0.3 B 0 0.4 0 0.4 C 0 0 0.5 0.5 D 0 0 0.6 0.6 ˜On the other hand, all of the vertices in G are strongly adjacent. Thus we can construct ¯ ˜the family Γ of fuzzy sets on V of G as in Table 3. ¯ ˜ Table 3: The Family Γ = {γ1 , γ2 , γ3 , γ4 } of the complement G Vertices γ1 γ2 γ3 γ4 Max A 0.3 0 0 0 0.3 B 0 0.4 0 0 0.4 C 0 0 0.5 0 0.5 D 0 0 0 0.6 0.6 ˜ ˜We can see that the chromatic number of G is χF (G) = 3 and the chromatic number ¯ ¯ ˜ ˜ ¯ ¯ ˜of its complement is χF (G) = 4 . Thus, χF (G) + χF (G) = 7 > 6 = 2(n − 1) . Thus, itsupper bound does not satisfy the Theorem 3.15. In the theorem 3.18 and 3.19, we add conditions for the theorem of Sattanathan andLavanya which is explained below.
10. 10 An Upper of fuzzy chromatic number of Fuzzy graphs and their Complement ˜ ¯ ˜Theorem 3.18. Let G(V σ, µ) be a fuzzy graph with n vertices and G = (¯ , µ) be a σ ¯ ˜ . If G satisﬁes the following properties:complement of G ˜ i. there is exactly one pair of vertices u, v ∈ V such that µ(u, v) = 1 min{σ(u), σ(v)} 2 ii. µ(x, y) = 1 min{σ(x), σ(y)} for all x, y ∈ V − {u, v} 2 ˜ ¯ ¯ ˜ then χF (G) + χF (G) = 2n − 1 and ˜ ¯ ¯ ˜ χF (G) · χF (G) = n(n − 1) ˜ ¯ ˜Proof. Let G(V σ, µ) be a fuzzy graph with n vertices and G = (¯ , µ) be a complement σ ¯ ˜ . Let V = {v1 , v2 , v3 , ..., vn }. Since G satisﬁes the following properties:of G ˜ i. there is exactly one pair of vertices u, v ∈ V such that µ(u, v) = 1 min{σ(u), σ(v)} 2 ii. µ(x, y) = 1 min{σ(x), σ(y)} for all x, y ∈ V − {u, v} 2 ˜then G has exactly one pair of vertices u, v ∈ V such that µ(u, v) > 1 min{σ(u), σ(v)} 2or µ(u, v) < 1 min{σ(u), σ(v)} . 2Without loss of generality, we assume v 1 and v2 are the vertices which satisfy µ(v1 , v2 ) <12 min{σ(v1 ), σ(v2 )} Then we can construct a fuzzy subset γ 1 of V where γ1 (v1 ) = σ(v1 ) , γ1 (v2 ) = σ(v2 ) ,and γ1 (vi ) = 0 for each vi ∈ V − {v1 , v2 } .On the other hand, other pair of vertices v i , vj ∈ V − {v1 , v2 } are strongly adjacent in G .˜Then we can construct fuzzy subsets γ k (k = 1, k = 2, 3, ..., n − 1) where γk (v) = σ(v) ifv = vk+1 and γk (v) = 0. if v = vk+1So we have the family Γ = {γ1 , γ2 , . . . , γn−1 } which satisfy the properies a,b,c in Deﬁnition ˜2.6. Thus χF (G) = n − 1 . ˜By theorem 2.7, since v1 and v2 are weakly adjacent in G then this vertices are stronglyadjacent in G ¯ . On the other hand, for each v , v ∈ V −{v , v } are also strongly adjacent ˜ i j 1 2in G¯ ˜ (By Theorem 2.5). So that every pair of vertices are strongly adjacent in G . By ¯ ˜ ¯ ¯ ˜Lemma 3.9, χF (G) = n . ˜ ¯ ¯ ˜Thus χF (G) + χF (G) = n + (n − 1) = 2n − 1 and ˜ ¯ ¯ ˜ χF (G) · χF (G) = n · (n − 1) .In the same way, if there is exactly one pair of vertices u, v ∈ V such that µ(u, v) >1 ˜2 min{σ(u), σ(v)} then we have χ F (G) = n and ¯ ¯ ˜ χF (G) = n − 1Theorem 3.19. Let ˜ ¯ ˜ G(V σ, µ) be a fuzzy graph with n vertices and G = (¯ , µ) be a σ ¯ ˜complement of G . If ˜ is not self complementary and G does not satisfy the properties G ˜in Theorem 3.18 then ˜ ¯ ¯ ˜ χF (G) + χF (G) ≤ 2(n − 1) and ˜ ¯ ¯ ˜ χF (G) · χF (G) ≤ (n − 1)2
11. I.Rosyida, S.Lavanya, Widodo, Ch.R.Indrati, K.A.Sugeng 11 ˜ ¯ ˜Proof. Let G(V, σ, µ) be a fuzzy graph with n vertices and G = (¯ , µ) be a complement σ ¯of G˜ . Since G is not self complementary and G does not satisfy the properties in ˜ ˜ ˜Theorem 3.18, then G has at least a pair of vertices which are weakly adjacent and G ˜ 1also has at least a pair of vertices u, v ∈ V such that µ(u, v) > 2 min{σ(u), σ(v)} . Since˜G has at least a pair of vertices which are weakly adjacent then by the proof of case 2 ˜ ˜of theorem 3.13 χF (G) ≤ n − 1 . On the other hand, since G has at least a pair ofvertices which are strongly adjacent then there is at least a pair of vertices which are ¯ ˜ ¯ ¯ ˜ ˜ ¯ ¯ ˜weakly adjacent in G . So χF (G) ≤ n − 1 . Thus χF (G) + χF (G) ≤ 2(n − 1) andχ (G) ¯ ¯ ˜ · χ (G) ≤ (n − 1)2 . ˜ F F 4. Conclusion In this paper, we give fuzzy chromatic number of a fuzzy cycle in Theorem 3.11 and3.12. We also give an upper bound for fuzzy chromatic number of a fuzzy cycle and itscomplement in Theorm 3.13. We also add conditions to a fuzzy graph so it satisﬁes thetheorem of Sattanathan and Lavanya as in Theorem 3.19. Further we are trying to ﬁndfuzzy chromatic number of other classes of fuzzy graphs, and fuzzy chromatic number ofoperations on fuzzy graphs. References [1] N. Biggs, Algebraic Graph Theory, Great Britain, 1974. [2] C.Eslahchi, B.N.Onagh, Vertex Strength of Fuzzy Graphs, International Journal of Mathematics and mathematical Sciences., 436 (2006) 1-9. [3] G.J. Klir, B.Yuan, Fuzzy sets and Fuzzy Logic-Theory and Applications, United State of America, 1995. [4] S.Lavanya, R.Sattanathan, Fuzzy vertex Coloring of Fuzzy Graphs, International review of Fuzzy Mathematics., Juli-December (2011) issue. [5] J,N Mordeson, P.S. Nair, Fuzzy Graphs and Fuzzy Hypergraphs, New York, 2000. [6] S.Munoz, M.T.ortuno, R.Javier and J.Yanez, Colouring Fuzzy Graphs, Omega:The Journal of Management Sciences., 33 (2005) 211-221. [7] MM.Pourpasha and M.R.Soheilifar, Fuzzy Chromatic Number and Fuzzy Deﬁning Number of Certain Fuzzy Graphs-Proceedings of 12th WSEAS International Confer- ence on Applied Mathematics, Cairo, 2007.
12. 12 An Upper of fuzzy chromatic number of Fuzzy graphs and their Complement [8] R.Sattanathan, S.Lavanya, Complementary Fuzzy Graphs and Fuzzy Chromatic Number, International Journal of Algorithms, Computing and Mathematics., 3 (2009) 21-25. [9] M.S.Sunitha, A.V.Kumar, Complement of Fuzzy Graph, Indian J. Pure Appl.Math., 33 (2002) 1451-1464.[10] L.A.Zadeh, Fuzzy Sets, Information and Control., 8 (1965) 338-353.
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